The VJ-Software Resolume Arena is a popular tool that is used by many media artists. The software allows you to create a composition of multiple video clips that are organized in individual layers.
Many tutorials that explain the various functionality of the software can be found on the website.
Projection Mapping
Resolume has a very powerful project mapping engine, that allows you to divide the composition into multiple slices. Later these slices can be mapped onto multiple surfaces. Additionally, Resolume also allows you to mask the mappings, if you for example want to project on a sphere or a more complex surface.